Beatty Sequences, Continued Fractions, and Certain Shift Operators

Abstract

AbstractLet θ = θ(k) be the positive root of θ2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.